The Stacks project

Proof. You can prove this theorem by carefully adjusting the arguments of the proof of Lemma 59.40.4 to the global setting. However, we want to indicate how we can glue the result of that lemma to get a global morphism due to the rigidity provided by the result of Lemma 59.40.2. Unfortunately, this is a bit messy.

Let us prove existence when $Y$ is affine. In this case choose an affine open covering $X = \bigcup U_ i$. For each $i$ the inclusion morphism $j_ i : U_ i \to X$ induces a morphism of locally ringed topoi $(j_{i, small}, j_{i, small}^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{i, {\acute{e}tale}}), \mathcal{O}_{U_ i}) \to (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X)$ by Lemma 59.40.1. We can compose this with $(g, g^\sharp )$ to obtain a morphism of locally ringed topoi

see Modules on Sites, Lemma 18.40.10. By Lemma 59.40.4 there exists a unique morphism of schemes $f_ i : U_ i \to Y$ and a $2$-isomorphism

Set $U_{i, i'} = U_ i \cap U_{i'}$, and denote $j_{i, i'} : U_{i, i'} \to U_ i$ the inclusion morphism. Since we have $j_ i \circ j_{i, i'} = j_{i'} \circ j_{i', i}$ we see that

Hence by uniqueness (see Lemma 59.40.3) we conclude that $f_ i \circ j_{i, i'} = f_{i'} \circ j_{i', i}$, in other words the morphisms of schemes $f_ i = f \circ j_ i$ are the restrictions of a global morphism of schemes $f : X \to Y$. Consider the diagram of $2$-isomorphisms (where we drop the components ${}^\sharp $ to ease the notation)

The notation $\star $ indicates horizontal composition, see Categories, Definition 4.29.1 in general and Sites, Section 7.36 for our particular case. By the result of Lemma 59.40.2 this diagram commutes. Hence for any sheaf $\mathcal{G}$ on $Y_{\acute{e}tale}$ the isomorphisms $t_ i : f_{small}^{-1}\mathcal{G}|_{U_ i} \to g^{-1}\mathcal{G}|_{U_ i}$ agree over $U_{i, i'}$ and we obtain a global isomorphism $t : f_{small}^{-1}\mathcal{G} \to g^{-1}\mathcal{G}$. It is clear that this isomorphism is functorial in $\mathcal{G}$ and is compatible with the maps $f_{small}^\sharp $ and $g^\sharp $ (because it is compatible with these maps locally). This proves the theorem in case $Y$ is affine.

In the general case, let $V \subset Y$ be an affine open. Then $h_ V$ is a subsheaf of the final sheaf $*$ on $Y_{\acute{e}tale}$. As $g$ is exact we see that $g^{-1}h_ V$ is a subsheaf of the final sheaf on $X_{\acute{e}tale}$. Hence by Lemma 59.31.1 there exists an open subscheme $W \subset X$ such that $g^{-1}h_ V = h_ W$. By Modules on Sites, Lemma 18.40.12 there exists a commutative diagram of morphisms of locally ringed topoi

where the horizontal arrows are the localization morphisms (induced by the inclusion morphisms $V \to Y$ and $W \to X$) and where $g'$ is induced from $g$. By the result of the preceding paragraph we obtain a morphism of schemes $f' : W \to V$ and a $2$-isomorphism $t : (f'_{small}, (f'_{small})^\sharp ) \to (g', (g')^\sharp )$. Exactly as before these morphisms $f'$ (for varying affine opens $V \subset Y$) agree on overlaps by uniqueness, so we get a morphism $f : X \to Y$. Moreover, the $2$-isomorphisms $t$ are compatible on overlaps by Lemma 59.40.2 again and we obtain a global $2$-isomorphism $(f_{small}, (f_{small})^\sharp ) \to (g, (g)^\sharp )$. as desired. Some details omitted. $\square$